Download On Linear Inference

On Linear Inference
Frank Pfennnig
Draft of February 2, 2008
Inference.
When we write an inference rule
even(x) true
even(s(s(x))) true
what do we mean? To discuss this, some terminology: we say “even(t)” is a
proposition and “even(t) true” is a judgment. Following Martin-Löf, a judgment
is an object of knowledge. We obtain knowledge by making inferences from
judgments we already know. We can then read the rule above as
If we know that t is even for a term t, we may conclude (and thereby
know) that s(s(t)) is also even.
The process of inference is therefore one by which we gain knowledge. We may
start with the knowledge that 0 is even, then we gain the information that
s(s(0)) is even, and next we learn that s(s(s(s(0)))) is even, and so on. This
process is clearly monotonic: we gain knowledge, but we never forget, at least
in the idealized world of mathematics.
Persistent and Ephemeral Truth. It has long been argued by philosophers
that truth is not universal, but depends on the world in which we consider a
proposition. This could depend on time (“It is raining now, but it did not rain
yesterday.”), place (“it is hot in the Sahara but cold at the North Pole.”) or
simply the state of a circumscribed system (“The white king is on square e1.”).
Studying logical inference in these situations is the domain of temporal, spatial,
or linear logic, respectively. In this note we discuss the latter.
We start by distinguishing two basic judgment forms, generalizing truth:
proposition A is persistently true (A pers) and A is ephemerally true (A eph).
Persistent truths are like the mathematical truths we have discussed earlier: the
number 2 is even, and this does not change. Ephemeral truths are those that
subject to change, those that depend on the current state. For example, after
we make a move, the white king may no longer be on square e1 and so this
proposition is ephemeral.
What is the nature of inference in the setting where we have persistent
and ephemeral truths? As an example, we consider the familiar blocks world
planning problems, with propositions on(x, y) (if x is on y), clear(x) (if x the top
of x is clear and x can therefore be picked up), free (the robot hand is free) and
holds(x) (the robot hand holds block x). We describe the possibility of picking
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up a block with the following rule:
on(x, y) eph
clear(x) eph
free eph
pickup
holds(x) eph
clear(y) eph
All judgments here concern ephemeral truth, and the rule can be read as follows:
If we are in a state where x is on y, x is clear, and the hand is free,
we can reach a new state where the hand holds x and y is clear.
Performing this inference means that the ephemeral propositions in the premiss
are no longer true in the new state, and the propositions in the conclusion are
now true in the new state.
An inference rule allows us to perform an inference, but does not force us to
do so. In the case of persistent (mathematical) truth, this means we may never
learn that 1592847498 is even. In the case of emphemeral truth, this means we
may never pick up a given block, even if the rule pickup would permit us to
do so. This is more important in this new setting because inferences may be
irreversible, so making an inference may constitute a real commitment. If all
truths are persistent (and hence inference is monotonic) we can always decide
to postpone application of a rule since it can be applied at any future point in
our deduction process.
New Forms of Rules. There is another new phenomenon, which is that
inference rule must be allowed to have not only multiple premisses, but also
multiple conclusions. The example above demonstrates this: we cannot break
pickup into two rules each with a single conclusion, not only because the premisses are consumed during the inference, but also because both conclusions
must appear in the state during the same atomic step. In our model of blocks
world, there is no intermediate state where y is clear but we do not yet hold x.
Similarly, it must be permissible for an inference rule to have no conclusion.
For example, if we had the possibility to discard a block that is on the table,
the rule might read
clear(x) eph
on(x, table) eph
discard
Mixing Persistent and Ephemeral Propositions. New considerations
arise when persistent and ephemeral propositions are combined. Consider that
blocks may either be small or large, and that the hand can only pick up small
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blocks. Since the size of the blocks does not change, small(x) is persistent.
small(x) pers
on(x, y) eph
clear(x) eph
free eph
pickup
holds(x) eph
clear(y) eph
The persistent nature of small(x) means that when a block x is picked up by
using this inference, the block remains small. Simple.
A slightly more subtle observation is that a persistently true proposition can
be used to satisfy the premiss of a rule A eph. We can think of the premiss A eph
as saying that A must be true in the state before the inference. Certainly,if A is
true in all states (that is, persistent) then it is true in any particular state where
we might need this fact. For example, we could represent a robot with more
hands than blocks with the persistent fact free. Now, any time during inference
we need to know that a hand is free, we can use this without destroying it.
Multiplicity. One final observation is that ephemeral truth can have a multiplicity. For example, we might have three nickels, represented by three judgments
nickel eph, nickel eph, nickel eph.
The rule for changes two nickels into a dime would be
nickel eph
nickel eph
dime eph
After applying this rule in the state above we have, as expected,
nickel eph, dime eph.
Which two of the three nickels we changed into a dime remains ambiguous, but
can be made precise if we label all components of a state with a unique label,
like a variable labeling each assumption in a deduction under the Curry-Howard
isomorphism.
In the whole discussion above, we only used atomic propositions, judgments
about them, and rules of inference. If we systematically construct a logic from
these considerations by internalizing the various notions, we obtain the judgmental formulation of intuitionistic linear logic by Chang et al. Relating propositions in linear logic back to the above formulation works only in a Horn-like
fragment, but is otherwise straightforward using the tools of focusing (see, for
example, the course notes on Logic Programming). We will do so now.
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A Horn-Like Fragment of Linear Logic. The fragment of linear logic we
consider allows us to express all of the above, now within a logic rather than
entirely at the level of judgments. This fragment is remiciscent of Horn logic,
but mixes ephemeral and persistent propositions.
Atomic propositions
Basic propositions
State propositions
State transitions
Persistent hypotheses
Ephemeral hypotheses
P
Q
S
R
Γ
∆
::=
::=
::=
::=
::=
P | !P
Q|1|S⊗S
S ( S | ∀x. R
· | Γ, R pers | Γ, P pers
· | ∆, R eph | ∆, P eph
The sequents defining the meaning of the connectives on this fragment have
the form
Γ; ∆ ` S eph
where Γ consists of persistent assumptions, ∆ consists of ephemeral assumptions, and S is the conclusion, a proposition describing a state. The inference
rules are the standard ones from the judgmental formulation of linear logic, so
we do not give them here.
The connection between reasoning in linear logic and the reconstruction
of persistent and ephemeral truth can be explored in both directions. In one
direction, alluded to above, we can internalize the judgment forms and obtain
linear logic with its full range of connective. In the other direction, we can
apply focusing to show that the inference rule formulation is indeed sound and
complete with respect to small step reasoning in linear logic. In order to do
this, we first present the focusing rules for linear logic on the fragment above.
All atomic propositions are considered positive, as are ⊗ and 1, while universal
quantification and linear implication are negative. We further restrict ∆ to
consist only of (positive) atoms. We obtain the rules below. Since the judgments
eph and pers can be inferred from the position in the sequent, we elide them
below to shorten the rules.
Focusing. First, from a neutral sequent Γ; ∆ ` S we can focus either on an
assumption R in Γ or on the right-hand side S. We cannot focus on something
in ∆ because it contains only positive atoms.
Γ; ∆; [R] ` S
R∈Γ
Γ; ∆ ` [S]
focusL!
Γ; ∆ ` S
Γ; ∆ ` S
focusR
Once we are focused on the left, we decompose the negative connectives, universal quantification and implication.
Γ; ∆; S(t) ` S
Γ; ∆; [∀x. R(x)] ` S
Γ; ∆1 ` [S1 ]
∀L
Γ; ∆2 ; [S2 ] ` S
Γ; (∆1 , ∆2 ); [S1 ( S2 ] ` S
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(L
When the left focus formula becomes positive, that is, a state formula S, we
transition to a left inversion phase.
Γ; ∆; S 0 ` S
Γ; ∆; [S 0 ] ` S
blurL
The left inversion phase will decompose ⊗ and 1 and move atomic formulas into
either Γ (if they are persistent) or ∆ (if they are ephemeral).
Γ; ∆; Ψ, S1 , S2 ` S
Γ; ∆; Ψ ` S
Γ; ∆; Ψ, 1 ` S
1L
Γ; ∆; Ψ, S1 ⊗ S2 ` S
(Γ, P ); ∆; Ψ ` S
Γ; ∆; Ψ, !P ` S
⊗L
Γ; (∆, P ); Ψ ` S
!L
atomL
Γ; ∆; Ψ, P ` S
When all negative connectives have been decomposed, we transition back to a
neutral sequent, ready to start another focusing phase.
Γ; ∆ ` S
Γ; ∆; · ` S
deactivateL
When we focus on the right, we decompose the state formula until we reach an
basic proposition. This must either be directly in the context (if it is emphemeral), or we lose focus and the underlying persistent propositions must follow
from the persistent hypotheses alone. This final sequent is also neutral.
Γ; ∆1 ` [S1 ]
Γ; · ` [1]
1R
Γ; ∆2 ` [S2 ]
Γ; ∆1 , ∆2 ` [S1 ⊗ S2 ]
⊗R
Γ; · ` P
Γ; P ` [P ]
A Prototypical Example.
atomR
Γ; · ` [!P ]
!R
As a prototypical example we consider the rule
A pers
B eph
C eph
D pers
where A, B, C, D, E are all atomic propositions. In linear logic, this rule can be
expressed as
!A ⊗ B ( C ⊗ !D.
Call this proposition R0 and add it to Γ, since the inference rules themselves
are persistent in the paradigm we are considering.
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What does it mean to focus on Ro , using the focusL! rule? Because a focused
derivations are quite restricted, the shape of the derivation is mostly determined.
At first, we just unwind R0 , leaving contexts and also the right-hand side indeterminate. We stop at initial sequents or neutral sequents.
, D; , C `
, D; , C; · `
, D; ; C `
; `A
; ` [!A]
!R
; ` [B]
; ; C, !D `
atomR
; ; C ⊗ !D `
⊗R
; ` [!A ⊗ B]
; ; [C ⊗ !D] `
deactivateL
atomL
!L
⊗L
blurL
(L
; ; [R0 ] `
R0 ∈
focusL!
; `
Filling in contexts, we notice first that the use of atomR requires B to be a
linear hypothesis. We further notice that the use of !R requires there to be no
ephemeral hypothesis in the subgoal proving A. Propagating this information
yields:
, D; , C `
, D; , C; · `
, D; ; C `
;· ` A
; · ` [!A]
!R
; B ` [B]
; B ` [!A ⊗ B]
; ; C, !D `
atomR
; ; C ⊗ !D `
⊗R
; ; [C ⊗ !D] `
deactivateL
atomL
!L
⊗L
blurL
(L
; , B; [R0 ] `
R0 ∈
focusL!
; ,B `
Next, we fill in the remaining holes parametrically and summarize the new
derived rule where premisses and conclusion are just neutral sequents.
Γ; · ` A
(Γ, D); (∆, C) ` S
R0 ∈ Γ
Γ; (∆, B) ` S
This gives a strong hint on how to read the view of inference from the introduction as a focused proof search strategy in linear logic. We think of Γ as
the persistently true propositions, ∆ the propositions ephemerally true in the
current state. We read the rule from the conclusion to the premise, as a step in
the search for a proof.
If we know the ephemeral proposition B, and the persistent proposition A follows from the all known persistent propositions Γ, then
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we can reach a state where D is persistently true and C is a new
ephemeral truth while rescinding B.
This reading does not quite coincide with our notion of linear inference discussed
first in this note. The mismatch is that arbitrarily complex reasoning can be
used to establish A, and this reasoning is to the side of our main inference
activity. This is clearly undesirable, because it means that the question if a
rule can be applied in a given state may be undecidable simply because of its
persistent premisses.
The analysis clearly shows that some additional restrictions are necessary
to bring inference with ephemeral and persistent truths in concordance with
focused linear logic.
Saturation. In preparation for the final step, we return to the setting where
all judgments are persistent truths. If we want to know if an atomic proposition
P follows from a known collection of atomic propositions and a set of rules, we
can simply keep applying rules when their premisses are satisfied. If we do this
in a fair manner (every rule that can be applied will eventually be applied), this
process is complete and if P is indeed true, it must eventually be derived.
We say the set of known atomic propositions is saturated if applying any
inference rule does not add any new knowledge. If our known facts are saturated,
then P follows precisely if we have already deduced it, so the test for P can be
reduced by a lookup.
In many cases we can present rules in such a way that the inference process
must always saturate. We then write Clo(Γ) for the closure of Γ under the
rules of inference. Moreoever, we can often determine the complexity of the
saturations process. It is evident from the foregoing that in such theories we
have Γ ` P if and only of P ∈ Clo(Γ).
Separation. Unfortunately, in the presence of ephemeral hypotheses and conclusions, saturation is not well-defined even if we start with no ephemeral truth.
For example, the rules
b eph
a pers
b eph
can generate and consume an arbitrary number of ephemeral b. We additional
restrictions to so that inference with persistent and ephemeral propositions truly
coincides with reasoning in linear logic. The analysis of focused reasoning suggests that we need a sufficient condition so that
Γ; · ` P
iff P ∈ Clo(Γ)
If we can achieve this, then the rule on the left, read in the forward direction
(from premisses to conclusion), corresponds to the focusing rule on the right,
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read in the backward direction (from conclusion to premisses).
A pers
B eph
A ∈ Clo(Γ)
C eph
D pers
(Γ, D); (∆, C) ` S
Γ; (∆, B) ` S
We would probably want to arrange things so that Γ is saturated with respect
to persistent propositions, so that the linear logic derived rule further simplifies,
just checking that A ∈ Γ.
Two straightforward conditions amount to a form of separation between
persistent and ephemeral reasoning
1. Every predicate symbol occurs only persistently or ephemerally. This
means no persistent proposition can realize an ephemeral premiss of a
rule.
2. Every rule with an ephemeral conclusion also has at least one ephemeral
premiss. This prevents us from circumventing the first restriction by introducing a new (renamed) propositions.
We conjecture that these are sufficient to guarantee that, on our fragment,
Γ; · ` P iff P ∈ Clo(Γ).
Range Restriction. So far we have concentrated on the propositional aspects
of inference, but the terms and parameters in the inference rules also present
some challenges. One difficulty can be see in a rule such as
positive(s(x))
which expresses that any number of the form s(t) for some t is positive. Since
there is no premiss, we can apply this inference for any t in place of x, but this
yields infinitely many different conclusions.
One way to resolve this to allow parametric truths to be asserted, rather
than just ground truths. This leads to what is traditionally called resolution,
where any clause is parametric in all of its free variables.
Saturation and complexity are easier to analyze if we restrict ourselves only
to ground truths. In that case, we may restrict rules to be range restricted which
means that every variable in the conclusion of the rule must also appear in at
least one premise. If the premisses are matched against ground truths (whether
ephemeral or persistent does not matter), then all variables in a rule will be
instantiated to ground terms, and the conclusion will also again be ground.
This level of choice is not represent in the focusing system in the form above,
but can easily be accomodated by replacing guessed terms (in the ∀L rule) by
meta-variables subject to unification (in the general case) or matching (in the
range-restricted case). A blueprint for this kind of generalization can be found
in the Logic Programming course notes.
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